MATH 433
Applied Algebra
Lecture 31:
Isomorphism of groups (continued).
The ISBN code.
Isomorphism of groups
Definition. Let G and H be groups. A function f : G → H
is called an isomorphism of the groups if it is bijective and
f (g1 g2 ) = f (g1 )f (g2 ) for all g1 , g2 ∈ G .
The group G is said to be isomorphic to H if there exists an
isomorphism f : G → H.
Theorem Isomorphism is an equivalence relation on the set
of all groups.
Theorem The following properties of groups are preserved
under isomorphisms:
• the number of elements,
• being Abelian,
• being cyclic,
• having a subgroup of a particular order,
• having an element of a particular order.
Examples of isomorphic groups
• (R, +) and (R+ , ×).
An isomorphism f : R → R+ is given by f (x) = e x .
• Any two cyclic groups hg i and hhi of the same order.
An isomorphism f : hg i → hhi is given by f (g n ) = hn for all
n ∈ Z.
• Z6 and Z3 × Z2 .
An isomorphism f : Z6 → Z3 × Z2 is given by
f ([a]6 ) = ([a]3 , [a]2 ).
• Z3 × Z2 and Z2 × Z3 .
An isomorphism f : Z3 × Z2 → Z2 × Z3 is given by
f ([a]3 , [b]2 ) = ([b]2 , [a]3 ).
Examples of non-isomorphic groups
• S(3) and Z7.
S(3) has order 6 while Z7 has order 7.
• S(3) and Z6.
Z6 is Abelian while S(3) is not.
• Z and Z × Z.
Z is cyclic while Z × Z is not.
• Z × Z and Q.
Z × Z is generated by two elements (1, 0) and (0, 1) while Q
is not generated by a finite set.
Examples of non-isomorphic groups
• Z × Z3 and Z × Z.
Z × Z3 has an element of finite order different from the
identity element, e.g., (0, [1]3 ), while Z × Z does not.
• Z8, Z4 × Z2 and Z2 × Z2 × Z2 .
It follows from the classification of finite Abelian groups.
• (R, +) and (R \ {0}, ×).
Suppose f : R \ {0} → R is a homomorphism. Then
f (−1) + f (−1) = f ((−1)(−1)) = f (1) = 0, hence
f (−1) = 0 = f (1).
Error-detecting/correcting codes
Messages sent over electronic and other channels are subject
to distortions of various sorts. Therefore it is important to
encode a message so that a possible error can be detected.
Then the receiver may ask that the message be repeated.
Such codes are called error-detecting.
To achive this, the message should carry a certain degree of
redundancy. One way to do this is a checksum. Namely, the
sender adds to a message one or several check symbols, which
are functions of the message. Then the receiver reevaluates
these additional symbols.
In some cases, requesting that the message be repeated is too
expensive. For such cases, we need a code that not only can
detect an error, but also allows to correct it. Such codes are
called error-correcting.
ISBN
International Standard Book Number (ISBN) is assigned
to all published books. It is an example of an error-detecting
code.
• ISBN-10 (old standard) consists of 9 decimal digits that
constitute the number followed by a check symbol, which is a
digit in base 11 (0–9 or X, the Roman notation for 10).
If a1 a2 . . . a9 a10 is the number, then
10a1 + 9a2 + 8a3 + · · · + 3a8 + 2a9 + a10
is to be divisible by 11. This happens for a unique choice of
a10 .
The code allows to detect one wrong digit or exchange of two
digits.
Example. 0 521 54050 X (ISBN-10 of the textbook).
ISBN
• ISBN-13 (new standard) consists of 13 decimal digits, the
last one being a checksum. If b1 b2 . . . b12 b13 is the number,
then b1 + 3b2 + b3 + 3b4 + · · · + 3b12 + b13 is to be divisible
by 10. This happens for a unique choice of b13 .
The code allows to detect one wrong digit or exchange of two
neighboring digits.
Old numbers are converted into new ones by adding 978 at
the beginning and recalculating the checksum.
Example. ISBN-10 of the textbook is 052154050X.
Therefore ISBN-13 of the textbook is 978-052154050d, where
9+3·7+8+3·0+5+3·2+1
+ 3 · 5 + 4 + 3 · 0 + 5 + 3 · 0 + d ≡ 0 mod 10.
We obtain that d = 6.